Finding Homogeneous Solutions for a Second Order Differential Equation

[Jalo]

Homework Statement

y''-4y'+4y=x*e^2x

I'm trying to find the homogenous solutions of this equation. I know there are two, but I can only find one.

YH=> y''-4y'+4y=0

Homework Equations

The Attempt at a Solution

y''-4y'+4y=0
Using the characteristic function:
a²-4a+4=a <=> (a-2)²=0
Therefore
C1*e^2x
is a solution. However, since this is a 2nd order differential equation, I should have two. I can solve the rest of the equation if I can find the 2nd solution. I've tought about it a lot but can't manage to find the answer... All help will be appreciated.
Thanks!

 

[Dick]

If you have a double root 2, then C2*x*e^(2x) is also a homogeneous solution.